Let’s say that a set contains an singular matrix if there exists an matrix with determinant zero whose entries are distinct elements of . For example, the set of prime numbers does not contain any singular matrices; however, for every it contains infinitely many singular matrices.
I don’t know of an elementary proof of the latter fact. By a 1939 theorem of van der Corput, the set of primes contains infinitely many progressions of length 3. (Much more recently, Green and Tao replaced 3 with an arbitrary .) If every row of a matrix begins with a 3-term arithmetic progression, the matrix is singular.
In the opposite direction, one may want to see an example of an infinite set that contains no singular matrices of any size. Here is one:
Continue reading Integer sets without singular matrices